Questions: Representation Ring

Question 1

In the representation ring R(G), what is the additive identity and what is the multiplicative identity?

Model answer: The additive identity is 0 (the zero virtual representation, corresponding to the zero-dimensional representation). The multiplicative identity is the class of the trivial representation [1_G], since V ⊗ 1_G ≅ V for any representation V.

R(G) is a commutative ring with the class [V ⊕ W] = [V] + [W] defining addition and [V ⊗ W] = [V]·[W] defining multiplication. The trivial representation acts as a tensor-product identity because V ⊗ ℂ ≅ V (where ℂ is the trivial representation). Virtual representations like [V] − [W] make sense formally even though negative-dimensional representations do not exist.

Question 2

For G = ℤ/nℤ with irreducible representations ρ₀, ρ₁, ..., ρₙ₋₁ (where ρₖ sends the generator to e^{2πik/n}), what is ρ_a · ρ_b in R(G)?

• A. ρ_{a+b} (with index mod n)
• B. ρ_a ⊕ ρ_b
• C. ρ_{ab} (with index mod n)
• D. The regular representation

Answer: (A)

Since ρ_a and ρ_b are both 1-dimensional, ρ_a ⊗ ρ_b is also 1-dimensional, with the generator acting by e^{2πia/n} · e^{2πib/n} = e^{2πi(a+b)/n}. This is ρ_{a+b mod n}. So R(ℤ/nℤ) ≅ ℤ[x]/(xⁿ − 1) as a ring, where x = [ρ₁]. The ring structure of R(G) captures all tensor product decomposition rules.

Question 3

A 'virtual representation' [V] − [W] in R(G) always corresponds to an actual representation of G.

• T. True
• F. False

Answer: False

Virtual representations are formal differences and do not correspond to actual representations when the subtracted part is nonzero. However, the character of a virtual representation (χ_V − χ_W) is a well-defined class function that can take negative values. Virtual representations are necessary to make R(G) a ring (not just a semiring) — without additive inverses, we could not do algebra. In topological K-theory, virtual bundles play an analogous role.

Question 4

The Adams operation ψᵏ on R(G) sends a representation V to a virtual representation whose character satisfies χ_{ψᵏ(V)}(g) = χ_V(gᵏ). What is ψ² of a 1-dimensional representation ρ?

Model answer: ψ²(ρ) = ρ², where ρ²(g) = ρ(g)². This is the representation obtained by squaring the character values.

For 1-dimensional ρ, χ_{ψ²(ρ)}(g) = χ_ρ(g²) = ρ(g²) = ρ(g)². So ψ²(ρ) = ρ ⊗ ρ = ρ². For higher-dimensional representations, ψᵏ is more subtle — it involves the power sum symmetric functions applied to the eigenvalues of ρ(g). Adams operations are ring homomorphisms (ψᵏ(V⊗W) = ψᵏ(V)⊗ψᵏ(W)) and play a central role in K-theory and the theory of λ-rings.